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Inventory Management V Finite Planning Horizon Lecture 9 ESD.260 Fall 2003 Caplice

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Assumptions: Basic FPH Model MIT Center for Transportation & Logistics - ESD © Chris Caplice, MIT Demand Constant vs Variable Known vs Random Continuous vs Discrete Lead time Instantaneous Constant or Variable (deterministic/stochastic) Dependence of items Independent Correlated Indentured Review Time Continuous vs Periodic Number of Echelons One vs Many Capacity / Resources Unlimited vs Limited Discounts None All Units or Incremental Excess Demand None All orders are backordered Lost orders Substitution Perishability None Uniform with time Planning Horizon Single Period Finite Period Infinite Number of Items One Many

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Example Demand Month When should I order and for how much? Costs D = 2000 items per year Co = $ per order Cp = $50.00 per item Ch = 24% per item per year Chp = ( Ch Cp )/ N = $1 per month per item N = number of periods per year More Assumptions Demand is required and consumed on first day of the period Holding costs are not charged on items used in that period Holding costs are charged for inventory ordered in advance of need MIT Center for Transportation & Logistics – ESD © Chris Caplice, MIT

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Five Basic Approaches 1. The One-Time Buy 2. Lot For Lot 3. Simple EOQ 4. The Silver Meal Algorithm 5. Optimal Procedures Wagner-Whitin (Dynamic Programming) Mixed Integer Programming MIT Center for Transportation & Logistics - ESD © Chris Caplice, MIT

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Approach: One-Time Buy On Hand Inventory Month MIT Center for Transportation & Logistics - ESD © Chris Caplice, MIT 2000

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Approach: One-Time Buy MIT Center for Transportation & Logistics - ESD © Chris Caplice, MIT MonthsDemand Order Quantity Holding Cost Ordering Cost Period Costs $1800$ $1650$ $1550$ $1500$0$ $1450$0$ $1300$0$ $1200$0$ $1000$0$ $800$0$ $550$0$ $250$0$ $0 Totals:2000 $13100$500$13600

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Approach: Lot for Lot On Hand Inventory MIT Center for Transportation & Logistics - ESD © Chris Caplice, MIT Month

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Approach: Lot for Lot MIT Center for Transportation & Logistics - ESD © Chris Caplice, MIT MonthDemand Ordering Quantity Holding Cost Ordering Cost Period Costs 1200 $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$500 Yotals:2000 $0$6000

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Approach: EOQ On Hand Inventory Month 400 MIT Center for Transportation & Logistics - ESD © Chris Caplice, MIT

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Approach: EOQ MIT Center for Transportation & Logistics - ESD © Chris Caplice, MIT Month Demand Order Quantity Holding Cost Ordering Cost Period Costs $200$500$ $50$0$ $350$500$ $300$0$ $250$0$ $150$0$ $ $200$500$ $ $150$500$ $250$500$ $0 Totals: 2000 $1900$2500$4400

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Approach: Silver-Meal Algorithm On Hand Inventory MIT Center for Transportation & Logistics - ESD © Chris Caplice, MIT Month

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Approach: Silver-Meal Algorithm MIT Center for Transportation & Logistics - ESD © Chris Caplice, MIT MonDmdLot Qty Order Cost Holding CostLot Cost Mean Cost 1 st Buy: 1200 $500$0$ $500$150$650$ $500 $150+$200 $850$ $500 $150+$200+$150 $1000$ $500 $150+$200+$150+$200 $1200$ $500 $150+$200+$150+$200+ $500 $1700$283 2 nd Buy: 6100 $ $500150$650$ $500 $150+$400 $1050$350

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Approach: Silver-Meal Algorithm MonDmd Lot Qty Order Cost Holding CostLot Cost Mean Cost 3 rd Buy: 8200 $500$0$ $500$200$700$ $500 $200+$500 $1200$400 4 th Buy: $500$0$ $500$300$800$ $500$300+$500$1300$433 5 th Buy: $500$0$500 MIT Center for Transportation & Logistics - ESD © Chris Caplice, MIT

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Approach: Silver-Meal Algorithm MonthsDemand Order Quantity Holding Cost Ordering Cost Period Costs $350$500$ $200$0$ $100$0$ $50$0$ $ $150$500$ $ $200$500$ $ $300$500$ $ $0$500 Totals:2000 $1350$2500$3850 MIT Center for Transportation & Logistics - ESD © Chris Caplice, MIT

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Approach: Optimization (MILP) MIT Center for Transportation & Logistics - ESD © Chris Caplice, MIT On Hand Inventory

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Approach: Optimization (MILP) MIT Center for Transportation & Logistics - ESD © Chris Caplice, MIT Decision Variables: Qi = Quantity purchased in period i Zi = Buy variable = 1 if Qi>0, =0 o.w. Bi = Beginning inventory for period I Ei = Ending inventory for period Data: Di = Demand per period, i = 1,,n Co = Ordering Cost Chp = Cost to Hold, $/unit/period M = a very large number…. MILP Model Objective Function: Minimize total relevant costs Subject To: Beginning inventory for period 1 = 0 Beginning and ending inventories must match Conservation of inventory within each period Nonnegativity for Q, B, E Binary for Z

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Objective Function Beginning & Ending Inventory Constraints Conservation of Inventory Constraints Ensures buys occur only if Q>0 Non-Negativity & Binary Constraints MIT Center for Transportation & Logistics - ESD © Chris Caplice, MIT

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Approach: Optimization (MILP) MIT Center for Transportation & Logistics - ESD © Chris Caplice, MIT

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Comparison of Approaches MIT Center for Transportation & Logistics - ESD © Chris Caplice, MIT Month DemandOTBL4LEOQS/MOPT Totals Cost $13,600$6,000$4,400$3,850$3,750

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